Bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank

Let G=(V,E) be a simple graph with vertex set V and edge set E. The rank of G, written as r, is defined to be the rank of its adjacency matrix. Let c denote e−v+θ, where e=|E|, v=|V| and θ means the number of connected components of G, and let m,α,χ′ respectively be the matching number, the independence number, and the chromatic index of G. In this paper, it is proved that ⌈r−c2⌉≤m≤⌊r+2c2⌋, ⌈2er+2c⌉≤χ′, and v−⌊r2⌋−c≤α≤v−⌈r2⌉. Examples are given to show that all the bounds can be attained.